compute a pre-comprehensive triangular decomposition
PreComprehensiveTriangularize(sys, d, R)
list of polynomials
number of parameters
The command PreComprehensiveTriangularize(sys, d, R) returns a pre-comprehensive triangular decomposition of sys, with respect to the last d variables of R.
A pre-comprehensive triangular decomposition is a refined triangular decomposition (in the Lazard sense) with additional properties, aiming at studying parametric polynomial systems.
Let U be the last d variables of R, which we regard as parameters. A finite set S of regular chains of R forms a pre-comprehensive triangular decomposition of F with respect to U, if for every parameter value u, there exists a subset S⁡u of S such that
(1) the regular chains of S⁡u specialize well at u, and
(2) after specialization at u, these chains form a triangular decomposition (in the Lazard sense) of the polynomial system F specialized at u. See the command DefiningSet for the term specialize well.
R ≔ PolynomialRing⁡x,y,s
R ≔ polynomial_ring
F ≔ s−y+1⁢x,s−x+1⁢y
A pre-comprehensive triangular decomposition of F consists of three regular chains.
pctd ≔ PreComprehensiveTriangularize⁡F,1,R
pctd ≔ regular_chain,regular_chain,regular_chain
Compare it with the output of Triangularize.
dec ≔ Triangularize⁡F,R,output=lazard
dec ≔ regular_chain,regular_chain
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